Applications of Transcendental Functions
D. P. Morstad, University of North Dakota
Objectives of Assignment
1.
To demonstrate characteristics and applications of the exponential
function f(x) = ex.
2.
To provide examples of logistic functions and applications which are
based on the exponential function.
AN IMPORTANT NOTE:
You might need to do a little research to find a formula required for this
assignment. The formula is not available in the Math Computer Lab, but it is
available in numerous math books and compendiums of mathematical
formulae – you might even want to look in your Calculus book!
I.
Basic Exponential Functions
All exponential functions have the unique and amazing property that the rate of change
of the function at any point is directly proportional to the value of the function at that point. That
is, the higher the function’s graph gets, the steeper the tangent line becomes. This is what
happens with compound interest and population growth.
For example, if you put $100.00 in a bank that pays 10% interest compounded
annually, your balance will grow exponentially. Although the interest rate is always 10%, the
dollar amount of the interest increases each year because the balance gets bigger. For example,
the first year you will only get $10.00 interest, but for the second year you will get more
because the balance will be $110.00. You will get $11.00 interest the second year. The third
year the interest will be $12.10. It’s always 10%, but 10% of bigger and bigger numbers. The
larger the balance, the faster it grows.
A similar thing happens with populations. The more animals or people or bacteria there
are, the bigger the population increase will be over a given time. If each generation is 10%
larger than the previous generation, the population will grow exactly like the bank balance in the
previous paragraph.
Sometimes, instead of increasing, quantities shrink at a rate proportional to the amount
of the quantity. This is what happens when radioactive elements decay. Strontium–90, for
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example, has a half–life of 28 years. This means that no matter how much Strontium–90 you
start with, it will always “shrink” by 50% in 28 years.
If the rate of change of a function is proportional to the value of the function, then
f ′( x ) = k ⋅ f ( x ) or y ′ = k ⋅ y , (where k is some constant).
Exponential functions are the only non–trivial functions that have this property.
Example 1. Graph and compare the rate of increase of the functions
f ( x ) = x , s( x ) = x 2 , g ( x ) = e 2( x −2 ) , and h( x ) = tan( x2 ) , on the
interval [0,4] .
Solution:
The window (0, 4, 0, 15) works quite well here. Graph all four of these
functions on the same graph. Be very careful with the parenthesis and keep track of which
function is graphed in which color for easy future reference. Then, just to provide some
bearings on the x–axis, type in “line(2,0,2,15)”. This draws a line from the point (2, 0) to the
point (2,15) so you will know where x = 2 is. Use online help to type e.
If you have given X(PLORE) all the right commands, your graph should look something
like this:
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s ( x)
f( x)
7.5
g( x)
h( x)
0
0
2
4
x
Although it is common to think that exponential functions grow extremely fast, notice
that g ( x ) = e2( x − 2 ) appears to be the most slowly growing function at first, and that it’s the
linear function which starts out with the steepest slope. Eventually, h( x ) = tan x2 appears to
()
have the steepest tangent lines. This should make sense since it has a vertical asymptote at x =
π while none of the other functions have vertical asymptotes. (This means that when you sketch
parabolas and exponential functions by hand, you should try to make them look as though they
are not asymptotic.)
Example 2. Interest on savings can be compounded annually, semi–annually, quarterly,
monthly, or as often as the bank decides. 8% compounded quarterly means 2% interest is paid
each quarter. 6% compounded monthly means 0.5% interest is paid each month. The more
frequently interest is compounded, the better deal it is for the depositor. The most lucrative
situation is when interest is compounded at every instant of time. This is referred to as
continuously compounded interest.
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The formula for finding the amount of money in such an account is A = Pe , where A
is the amount, P is the original deposit or principal, r is the annual interest rate, and t is the
number of years the money has been in the account.
Suppose $1,000.00 is deposited in an account that pays 8% interest compounded
continuously, and $1,500.00 is deposited in another account that pays 5% compounded
continuously. How long will it be before the 8% account is worth more than the 5% account?
rt
Solution:
Graph f ( x ) = 1000e 0.08 x and g ( x ) = 1500e0.05 x . Using the crosshairs,
you can see that it takes about 13.5 years. (This problem could also be solved algebraically.)
II. Logistic Curves
In the real world, population growth cannot truly be exponential. This is due to
limitations of space, resources, and the effects of disease. During the first stages of growth, a
population may appear to grow exponentially, but there will be a limit to how many members of
the population the environment can support. This limiting number is called the carrying
capacity of the environment.
As a population approaches the carrying capacity, its growth slows and it begins to
approach the carrying capacity asymptotically. In some cases, which we will not examine here,
once a population reaches a sufficient level of density, disease or famine can suddenly play a
devastating role and cause the population to plummet drastically. The stability of a population
might be tenuous at best ,and the population might oscillate between two or more specific levels.
(This is reflected in periodic plagues of locusts and other insects. This is not to be confused with
cycles due to lengthy larval stages.) We will only consider stable growth.
In higher level applied math classes you might learn that this type of bounded stable
growth follows what is called a logistic curve. Logistic curves accurately model the stable
growth of most things in the real world: growth of populations, rate of learning, growth of
businesses, etc. A very simplistic version of the logistic curve is given by:
f (t) =
P
, where P is the carrying capacity and a is a growth rate.
1 + e−at
If you want to invest in a rapidly growing business or determine how likely a population
will continue to grow at a rapid pace, you need to determine its position on its logistic curve.
Example 3. On the same axis system, graph the logistic curve for each of the following:
a) P = 4, a = 2; b) P = 3, a = 2; c) P = 3, a = –2; d) P = 3, a = 4.
How does a change in P affect the logistic curve? How does the value of a affect
the curve?
Solution: Choose an appropriate window (don’t ask the teacher – figure one out by yourself
by thinking or by using trial and error), and graph all four curves. Analyzing the curves, P is the
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non–zero horizontal asymptote, and a influences how long the period of fast growth rate lasts
(or the decay lasts if a < 0). Larger values of a result in a shorter period of fast growth, but in
faster growth during this “spurt” period.
Example 4. Country X’s population grows according to a logistic curve with P = 107
and a = 10, and country Y’s population grows with P = 108 and a = 1, where t is
in decades. Which country reaches 5,000,000 first?
Solution: Think in terms of ten millions. Then the P’s are 1 and 10 respectively, and you want
to find which gets to 0.5 first. Graph both using a good window. The crosshairs allow you to
determine that country Y wins. It reached 5,000,000 about 3 decades ago.
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